# MODELING AND SIMULATION OF A THREE-PHASE INVERTER WITH RECTIFIER-TYPE NONLINEAR LOADS

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2 voltage is proposed in [1]. he technique also includes a load predictive feed-forward loop in a voltage controller and an output voltage feed forward loop in a current controller. In [17], linear and nonlinear adaptive control strategies for three-phase UPS inverters have been presented. An on-line adaptive learning algorithm has been also described which promotes steady state controller stability. o prove the effectiveness of the proposed technique, various simulation results using Matlab/Simulink are shown under both no-load and nonlinear loads. he remainder of this paper can be outlined as follows. In section II, a mathematical model of the proposed system including three-phase PWM inverter, nonlinear load, control system and multiple-filter is described. In section III, a small signal equivalent circuit and transfer function dq model of the system are derived. Simulation results are reported and discussed in section I. Finally, a brief summary is given in section. 2. Mathematical model of the system Analytical model is an important tool for prediction of dynamic performance and stability limits using different control laws and system parameters. Mathematical model of the inverter must be established before the design stage. A schematic diagram of the proposed three-phase UPS inverter system is shown in Fig.1, it mainly consists of a controller, switching bridge and an output filter. he block diagram of the system can be divided into four parts: 2.1. A hree-phase PWM Inverter Fig. 2 shows a typical configuration of a three-phase full-bridge UPS inverter. If switching frequency is high enough, the PWM inverter is considered as a voltage source inverter (SI) and dynamic response of the UPS inverter is mainly determined by the elements of the filter. SPWM techniques are applied to inverters in order to obtain a sinusoidal output voltage with minimal undesired harmonics. Semiconductor switching devices (Q1 Q) of the inverter are controlled by PWM signals to obtain three-phase near sinusoidal ac voltages of the desired magnitude and frequency at the inverter output. he operation of three-phase inverter can be defined in eight modes as shown in able I which shows status of each switch in each operation mode. hree-phase switching state functions S a, S b and S c of the inverter are used to calculate the line output voltages of PWM inverter which is described by the help of Fig. 2 [18]: where [ ] [ ] U U U = U AB BC CA A B C dc (1) 8

3 A 1 1 Sa 1 B 1 1 S = b. 2 C 1 1 S c (2) Fig.1. Structure of a three phase inverter system. Fig.2. Basic three-phase voltage source inverter. able 1. Reference frame dq voltage. Mode qs ds os A dc 2 B dc dc dc C dc dc dc D 2 dc E 2 F G dc H 2 9

4 he inverter input current (I IN ) can be calculated as: Armenian Journal of Physics, 29, vol. 2, issue 4 IIN = SAIAF + SB IBF + SC ICF. () Fig. shows the block diagram detailed description for inverter phase and line voltages based on the transfer functions. he switching functions can be mathematically represented as follows [19]: Sa( ω t) = Ak sin kωt, k = 1,,... 2π Sb( ω t) = Ak sin k( ωt ), k = 1,,... 4π Sc( ω t) = Ak sin k( ωt ). k = 1,,... (4) 2.2. Nonlinear Load Nonlinear loads draw non-sinusoidal current, even when connected to a sinusoidal voltage. Also, the voltage and current waveforms are not of the same shape and contain fundamental frequency as well as non-fundamental frequencies. hree-phase loads that use three-phase rectifier in distribution systems include adjustable motors, UPS systems and battery charger. When a nonlinear load such as a capacitor input diode bridge is connected to UPS, the peaky current flows once in a half cycle and the output voltage become distorted. hree-phase rectifier is used for higher power applications, up to several MW. he circuit of the full-bridge rectifier for three-phase system is shown in Fig.4. he rectifying topology is similar to single-phase rectifiers but with a front end for connection of three-phase. he non-sinusoidal load current is drawn from the mains. If the diode bridge is turned on, the nonlinear load can be expressed as d 1 1 i, L = u O + u LINE (5) dt L L DC DC d 1 1 u, O = i L i O () dt C C DC where u O is the dc-side voltage of diode bridge rectifier, i L is the nonlinear load current, L DC is the inductor and C DC is the capacitor dc-side of LC filter. When the diode bridge is turned off, the nonlinear load can be expressed as d u dt DC 1 = i (7) C O O, Eqs. (5) (8) define the dynamic model for non-linear load. DC i =. (8) L 1

5 Fig.. Block diagram based on switching state functions for Inverter line and phase voltages 2.. Multiple-filter Fig4. Nonlinear load. Some harmonics success to pass through the load when a mono-filter is used. he harmonics can be more reduced by going for multiple-filter. A multiple-filter is obtained by connecting a LC filter between the first filter and the load. Output multiple-filter connects the utility (A, B and C) inverter to the load (X, Y and Z) through two LC filters, as shown in Fig.5. For a balanced threephase system, the following set of equations describes the voltage and current conditions on the multiple-filter: d RF 1 1 i = i u + u ABC, (9) dt L L L F F F d 1 1 u, = i i (1) dt C C F F d RS 1 1 i = i u + u, (11) dt L L L S S F d 1 1 u, = i i LABC (12) dt C C S where u is the first filter capacitors voltage, u is the second filter capacitors voltage, i is the inverter output currents, i load current: S is the second filter inductors current and i LABC is the three-phase 11

6 [ ], u = u u u (1) FA FB FC [ ], u = u u u (14) SA SB SC [ ], i = i i i (15) FA FB FC [ ], i = i i i (1) SA SB SC [ ], i = i i i (17) LABC LA LB LC Fig.5. Inverter output filter. Also, L F and L S are the filter inductances, C F and C S are filter capacitances and R F and R S are the equivalent series resistances of the filter Control system In most cases, control design for a three-phase PWM converter involves two steps: choice of modulation strategy, which corresponds to an open-loop converter control, and design of dynamic closed loop control. he instantaneous voltage control scheme is applied to the proposed circuit. Fig. shows the block diagram of the control circuit. he output voltage is compared with a reference sinusoidal wave R. Hence, the control scheme is simple compared with others. he classic PID (proportional integral-derivative) controller has been used in many industrial control systems, mainly due to its simple structure that can be easily understood and implemented in practice, and its excellent flexibility made possible by adjustment of the coefficients K P, K I and K D.. DQ model of small signal analysis Small signal model can obtained using perturbation and linearization around an operation point. he dynamics of some converter variables can be analyzed using small signal model. An open-loop dynamic analysis has been made using the linear small signal model. his analysis is based on transfer functions and Bode plots, and is similar to the analysis of single-input-singleoutput systems. u Fqd i and i LABC, u Sqd, respectively,, i Fqd, i Sqd and i Lqd are the qd components of the u, u, i, 12

7 u = u u, (18) Fqd Fq Fd u = u u, (19) Sqd Sq Sd i = i i, (2) Fqd Fq Fd i = i i, (21) Sqd Sq Sd i = i i. (22) Lqd Lq Ld he dq rotating reference frame model of a three-phase UPS inverter with multiple-filter output is obtained using (9)-(12). he q and d sub-circuits have coupled voltage and current sources. ransfer functions for the various feedback control schemes are used to compare their anticipated performance characteristics, before proceeding to detailed simulation. he channels must be viewed a multiple input and multiple output (MIMO) system, so the system has the four following transfer functions: H H dd qd UFd () s () s =, U () s (2) ds UFq () s () s =, U () s (24) ds UFq () s Hqq () s = Hdd (), s U () s = (25) qs Fd () s Hdq () s = Hqd (). s () s = (2) qs herefore, all four transfer functions have the same resonance frequency and damping factor. By ignoring coupling of the power stages, the cross coupling transfer function in H qd (s) and H dq (s) would be zero. Equations system can be represented by equivalent circuits shown in Fig. 7. he equivalent series resistances of the filters capacitors are not considered in the model. 4. Simulation results In power electronic systems, simulations are mainly performed to analyze and design the circuit configuration and applied control strategy. A three-phase UPS has been simulated to verify the operation of the proposed control method. Fig.8 shows the simulation results including load current, load voltage and modulation index, under three-phase inductive load. Fig.9 show the simulated waveforms of the output voltage and output current of the system with non-linear load when multiple LC output filter is employed. he HD of the voltage is less than 5% and the regulation is less than %. 1

8 Fig.. Equivalent dq model of three-phase UPS inverter with multiple filters. Fig.7. Block diagram of dq rotating frame controller (a) Load current (b) Load voltage (c) Modulation index Fig.8. Simulation results under three-phase inductive load. 14

9 Fig.9. oltage and current of the non-linear load. 5. Conclusion A mathematical modeling of a system for analysis and design of the proposed control system was described. he mechanism of output waveform distortion of three-phase PWM inverter with nonlinear loads through detail theoretical analysis was explained. o eliminate the harmonic components transferred to the load, a multiple-filter was used at the output of UPS. his filter enables to reduce HD of the output voltage about 5% and voltage regulation %. References 1. J.Faiz, G.Shahgholian, Electomotion, 1(4), 27 (2). 2. R.L.A.Riberio, F.Profumo, C.B.Jacobina, G.Griva, E.R.C.D.Silva, A.M.N.Lima, IEEE/IAS, 2, p 89 (22).. P.Zhu, X.Li, Y.Kang, J.Chen, IEEE/IPEMC, 2, 9 (24). 4. J.Faiz, G.Shahgholian, M.Ehsan, Euro. rans. Electr. Power, 18, 29 (28). 5. K.Zhou, K.Low, D.Wang, F.Luo, B.Zhang, Y.Wang, IEEE rans. on Pow. Elect., 21(1), p. 19 (2).. G.Escobar, A.A.aldez, J.L.Ramos, P.Mattavelli, IEEE rans. On Indu. Elec., 54(1), 54 (27). 7. G.enkataramanan, D.M.Divan,.M.Jahns, IEEE rans. On Pow. Elec., 8(), 279 (199). 8. S.Buso, S.Fasolo, L.Malesani, P.Mattavell, IEEE rans. On Indu. Appl., (4), J.S.Cho, S.Y.Lee, H.S.Mok, G.H.Choe, IEEE/IAS, 4, S.Buso, S.Fasolo, P.Mattaveli, IEEE rans. On Indu. Appl., 7(), P.C.Loh, M.J.Newman, D.N.Zmood, D.G.Holmes, IEEE rans. on Pow. Elec., 18(5), 117 (2). 12. G.Willmann, D.F.Coutinho, L.F.A.Pereira, F.B.Líbano, IEEE ran. on Ind. Elec., 54(), 1591 (27). 1..Kawabata,.Miyashita, Y.Yamamoto, IEEE rans. on Pow. Elec., 5(1), 21, O.Kukrer, IEEE rans. on Pow. Elec., 11(1), J.Faiz, Gh.Shahgholian, ECI-CON,

10 1. F.S.Pai, S.J.Huang, IEEE rans. on Pow. Elec., 21(1), J.H.Choi, B.J.Kim, IEEE/APEC, 2, L.G.Barnes, R.KrishnanIEEE/PESC, 1, 47 (1995). 19. B.K.Lee, M.Ehsani, IEEE rans. on Indus. Electr., 48(2), 9 (21). 2. J.Faiz, Gh.Shahgholian, ICEMS, Jeju Island, Korea, November 24. 1

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